Consistency of Imperfections in Steel Eurocodes

: The second-order theory was used to analyze the flexural buckling of an individual member simply supported on both member ends, with a uniform double symmetric cross-section under a uniform axial force in an elastic state. The purpose was to show the influence of four different amplitudes of initial imperfections on the shape of the elastic buckling mode 𝜂 (cid:2913)(cid:2928) (𝑥) used in the current EN 1993-1-1 and its new draft, prEN 1993-1-1. Three methods were followed for the analysis: the equivalent member (EM) method, the unique global and local initial (UGLI) imperfection method, and second-order theory with the initial imperfection having an initial local bow imperfection 𝑒 (cid:2868) . For the relevant quantities, simple formulae were derived and their distribution was drawn on diagrams to represent their graphical interpretations for the first time ever. The formulae and diagrams were valid for the ultimate limit state, which means 𝑁 (cid:2889)(cid:2914) = 𝑁 (cid:2912),(cid:2902)(cid:2914) . The influence of four different amplitude values was evaluated: (a) 𝑒 (cid:2868),(cid:2921) , proposed for the UGLI imperfection method in the draft EN 1993-1-1; (b) the initial local bow imperfection 𝑒 (cid:2868) , utilized in the current EN 1993-1-1; (c) the other one employed in its draft; and (d) 𝑒 (cid:2868),(cid:2914) , used in the UGLI imperfection method in the current EN 1993-1-1, the current EN 1999-1-1, and the draft prEN 1999-1-1. The main conclusion was that 𝑒 (cid:2868),(cid:2921) must not be used in the draft EN 1993-1-1. The UGLI imperfection method was also applied to the column fixed at one end and simply supported at the other end. This example showed the geometrical interpretation of relevant amplitudes. The historical development of the UGLI imperfection method is also presented. All the relations are illustrated in two numerical examples, and the geometrical interpretations of formulae were used in the diagrams. The partial results were verified by the independent computer programs FE-STAB and IQ 100.


Introduction
The causes of imperfections of members may include the following: (a) a deviation from straightness and twist; (b) unavoidable eccentricity due to variation in the crosssectional dimension; (c) eccentricity due to non-centric loads; (d) a residual stress pattern; etc. All these causes can be explained by an equivalent geometrical imperfection, which is defined by its shape and amplitude.
The shape of the equivalent geometrical imperfection in EN 1993 (Part 1-1) was chosen as the elastic buckling mode ( ) of the structure, i.e., for the simply supported column ( )sin( ⁄ ), because then ( ) and the deformation from the compression force would be affine. This fact simplifies both the calculation and result. The bending moment = , is only of interest at the midspan of the column because the compression force and the cross-section are uniform along the column length.
Four different amplitudes appear in EN 1993 (Part 1-1): (a) amplitude , , hidden in the reduction factor of the equivalent member (EM) method; (b) an initial local bow imperfection in the current EN 1993-1-1; (c) the other one in the new draft, prEN 1993-1-1; and (d) amplitude , , used in the UGLI imperfection method.
To show the differences caused by the various amplitudes used in the different methods, a decision was made to investigate individual members simply supported on both member ends with uniform double symmetric cross-sections under a uniform axial force in an elastic state.
The following methods show how the stability of an individual member under compression can be checked: (a) The EM method; (b) The UGLI imperfection method, which is the second-order analysis of a member under compression with a unique global and local initial (UGLI) imperfection; (c) The second-order analysis of a member under compression, with a shape imperfection that is derived from the elastic buckling mode of the structure with an amplitude of the equivalent geometrical bow imperfection.
All these methods are described. For the relevant quantities, formulae are given together with their geometrical interpretations, which may be useful for designers and educational institutes. The old and new Eurocode initial local bow imperfections were compared to one another, and also to the , and , values. It was shown that the decision made in the new prEN 1993-1-1, namely to use , in the UGLI imperfection method (in fact, safety factor was removed from the , used in the current EN 1993-1-1), has important negative consequences. It was also proven that the procedure followed in the current EN 1993-1-1, EN 1999-1-1, and the draft prEN 1999-1-1, and that used in the UGLI imperfection method, gives an , > , value for > 1.0 that is correct.

Equivalent Member Method
The derivation of the reduction factor χ for the relevant buckling curve: The basic formula for the flexural buckling curve to determine the characteristic resistance of a member under compression reads as so: After using the characteristic values of cross-section resistances: then Equation (1) may be rewritten as follows: The factor by which the design value of the loading would have to be increased to cause elastic instability is as follows: The amplification factor after taking into account second-order effects is as follows: The bending moment at the member midspan according to the first-order theory is as follows: The bending moment at the member midspan according to the second-order theory is as follows: The characteristic imperfection amplitude value is as follows: The shape of the equivalent geometrical initial imperfection ( ) is that of the structure's elastic critical buckling mode ( ). For the investigated member, it comes in the form of a sinus half-wave and includes both structural and geometrical imperfections, as shown by the following: The imperfection factors corresponding to the appropriate buckling curves are shown in Table 1, together with = 0, which is valid for the ideal member.
After inserting the reduction factor χ (11) and fraction (12) from Equation (3), the following was obtained: The basic quadratic Equation (13) was obtained for reduction factor χ: which may be rewritten in the following way: The solution of quadratic Equation (14) leads to "European column buckling curves" (Figure 1). A compression member should be verified against buckling as follows: Equation (15) may be written in this way by showing the utilization factor value: The maximum design value of the compression force equals the design buckling resistance of the compression member , when the utilization factor = 1.0: The same results for = , can be obtained when the second-order theory is used under the condition that the characteristic value of the imperfection amplitude , is replaced with the design value of the imperfection amplitude , : The relative design value of the imperfection amplitude is as follows: where: The following equations are valid for the maximum design value of the compression force = , when the utilization factor = 1.0: Consequently, the following were similarly obtained as above: The design value of the amplitude , , of the additional deformation ( ) caused by the force on the member with the equivalent geometrical initial imperfection ( ) is as follows: The relative design amplitude , , ⁄ of the additional deformation , ( ) is as follows: , , = , = . (28) The amplitude , of the total deformation ( ) is as follows: The relative amplitude , ⁄ of the total deformation ( ) is as follows: The partial utility factors are: The utility factor is as follows: Evidence for the utility factor , as defined by Equation (32), equaling 1.0 may be obtained after inserting the relative slenderness (Equation (10)) and reduction factor (Equation (13)) in Equation (32) and arranging it. This may be achieved, e.g., by the MATHCAD commands Symbolics and Simplify.
Removing safety factor from amplitude , leads to its value being 1.00 ÷ 2.19 times lower (Table 2).

Numerical Example 1
Comparison of the verifications according to Equivalent Member method and second-order theory is illustrated in Figure 2.
The results of the second-order theory are summarized in Table 3.
If the calculation in Equation (34) is repeated according to the second-order theory with , instead of , (that is, when is removed from , ), the following results are obtained: , , , , = = 6.798 mm, In this case, removing the safety factor from the amplitude , leads to an incorrect value of the utility factor , which is calculated by the second-order theory, where 0.5% is on the unsafe side.
The results of the second-order theory are summarized in Table 4.
In this case, removing the safety factor from the amplitude , leads to an incorrect value of the utility factor , which is calculated by the second-order theory where 15% is on the unsafe side. The

Geometrical Interpretations of the above Quantities
The quantities           The diagrams enable the determination of: (a) the amplitudes of all kinds of deformations, and (b) the influence of the second-order theory. The first-order analysis may be used for the structure if the increase in the relevant internal forces or moments, or any other change in structural behavior caused by deformations, can be neglected. This condition may be assumed fulfilled if the following criterion is met:  (45)). The FE-STAB results confirm the correctness of the values in Table 4

Historical Development of the UGLI Imperfection Method
Chladný prepared Table B.1 for STN 73 140:1998 partly based on [2] to determine the imperfection equivalent of a member under compression, for which he calculated and proposed the values of relevant parameters based on his own study. The equivalent imperfection form corresponds to pre-standard ENV Eurocode 3 [2], which differs from the later EN Eurocode 3 drafts [3,4] and the current Eurocodes 3 and 9.
In October 2000, Chladný sent to Prof. J. Brozzetti comments on [2] and on [3] regarding the equivalent imperfection value of a member under compression. In the draft prEN Eurocode [3], the imperfection form corresponded to the pre-standard ENV Eurocode [2]. Consequently, the imperfection form was changed in the later prEN Eurocode 3 drafts. See [5] as well.
The list of references in [5] cites the work of Chladný E. (item [52] in [5]) as the only one from countries of the former Central and Eastern Europe and non-CEN members.
Until February 2004, the working draft prEN 1999-1-1 [6] did not contain provisions concerning the shape and value of a single equivalent for the global and local imperfections of a member under compression. For prEN 1999-1-1 [6], Baláž and Chladný also prepared a proposal for the clause containing the shape and amplitude of the equivalent geometrical UGLI imperfection of a member under compression. Chladný developed the proposal for EN 1999-1-1, which differs from that which was accepted in EN 1993-1-1 [7]. Baláž modified Chladný's proposal by adding the absolute values and other changes used in the formulae in such a way that the formulae gave correct numerical results. In EN 1993-1-1 and in 5.3.2(11), instructions are given for determining the amplitude of the UGLI imperfection of compressed structures with a cross-section and a constant axial force along their length. In EN 1999-1-1 [8], the more general procedure is given in 5.3.2 (11), which is also valid for members with a variable and/or non-uniform axial force. Baláž and Chladný sent their more general procedure to TC 250/SC9. Their proposal was accepted by prof. Torsten Höglund in Sweden, who was the head of Project Team PT Members in Subcommittee SC9, and by prof. Federico Mazzollani in Italy, who was the chairman of Subcommittee TC 250/SC9. The procedure can be found in the working drafts starting with draft prEN 1999-1-1, April 2004. The more general procedure by Baláž and Chladný was also accepted in the Slovak National Annex [9].
The above historical development has been described in detail by Baláž in the proceedings [10], where in Chapter 5, Chladný published his theory for the first time together with detailed numerical examples and applications in many areas. Chladný gave presentations in 2007 in courses for practicing designers and for people from universities. In the second edition of the proceedings [11], Baláž and Chladný corrected Chapter 5. In the courses for practicing designers and for people from universities, the 2010 presentations were given by Baláž because Chladný was ill.
Baláž derived Chladný`s method differently [12,13] and showed that the UGLI imperfection and its amplitude have a geometrical interpretation, which is helpful for practicing designers. The geometrical interpretation is valid only for structures with a crosssection and a constant axial force along their length.
Baláž asked Chladný to publish his theory in English. The UGLI imperfection method was published by Chladný and his daughter in English in [14][15][16], where more details about the method are found. Baláž`s contributions to the UGLI imperfection method are acknowledged by Chladný at the end of [15]. Baláž [7], also influences the value of the amplitude of UGLI imperfection. These reasons cannot be accepted at all. The opposite is true, which is clearly explained in this paper.
A detailed explanation of the UGLI imperfection method and its geometrical interpretation can be found in [19].
An application of the UGLI imperfection method on the large Žďákov steel arch bridge is published in [20].

UGLI Imperfection Method Proposed by Baláž and Chladný for the Drafts of Eurocodes [17] and [18]
As an alternative to the second-order theory with the amplitude of the equivalent bow imperfection given in cl.
where the design value of , is given by: where denotes the critical cross-section of either the frame structure or the verified member (see Note 5); is the index that indicates belonging to the critical cross-section; is the imperfection factor for the relevant buckling curve; = , , , is the relative slenderness of either the frame structure or the verified member and of the equivalent member (see Note 2); is the limit of the horizontal plateau for the relevant buckling curve; is the reduction factor for the relevant buckling curve and slenderness is the value of the axial force in the critical cross-section, , when the elastic critical buckling is reached, and it is also the critical axial force for the equivalent member; is the minimum force amplifier for the axial force configuration in members to reach the structure's elastic critical buckling; , , is the characteristic moment resistance of the cross-section, ; , , is the characteristic normal force resistance of the cross-section, ; and ´´( ) is the value in the critical cross-section, , of the bending moments, which would be necessary to bend the structure (in the state without axial forces) in the form of the buckling mode.
Note 1: Formula (46) is based on the requirement that an imperfection, , ( ), in the shape of the elastic buckling mode, ( ), should have the same maximum curvature as assumed for the equivalent member method in ( ≤ , ). Therefore, the buckling resistance of the uniform members loaded in axial compression, and calculated with the imperfection according to (46) and for second-order effects, is identical with the value of , = . ⁄ The imperfection, , ( ), in the shape of the elastic critical buckling mode is generally applicable to all members under compression and to frames buckling in their plane. It is especially suitable for members with cross-sectional characteristics and/or an axial force that is not constant along their length, and also for frames containing such members.
Note 2: The equivalent member has pinned ends, and its cross-section and axial force are the same as in the critical cross-section, , of the frame. Its length is such that the critical force equals the axial force in the critical cross-section, , upon the critical loading of the structure. Note 3: To calculate the amplifier, , the members of the structure may be considered to be loaded by axial forces, , from the first-order elastic analysis of the structure. Note 4: The formula Effect of , ( ) imperfection. Deflexion ( ), as the effect of the imperfection ( ) on the compressed member, is calculated using the second-order analysis as follows [14]: where generally, for a non-uniform member with non-uniform distribution of axial force: The maximum of the equivalent geometrical initial imperfection , ( ) is given by:  (55)
The buckling perpendicular to axis z-z of the column simply supported at both ends was investigated, which was solved in Numerical Example 1 by two other methods. The input values were taken from Numerical Example 1. The axial force = , = 859.584 kN acted on the column with length = 12 m and simply supported at both member ends, with a uniform cross-section IPE 500 made from S235 steel. For this case, the calculation according to the UGLI imperfection method gave identical results as the calculation according to the second-order theory in Table 4. That is why all the steps of the calculation according to the UGLI imperfection method described above were applied for the more general case, in which the same column was fixed at the bottom and its upper end was simply supported. See the next numerical example, 2b.
The relative slenderness is given by: For the buckling curve "a", the imperfection factor = 0.34.
The eigenfunction of the buckling mode and its first derivation are given by: The maximum of function ( ) in section = 7.22 m, which was found from the condition ´( ) = 0. The amplitude of ( ) is = , = ( ) = 6.2832. It was not necessary. Nevertheless, below the normalized eigenfunction, it was used with an amplitude equaling 1.0: The amplitude of the additional deformation ( ) is given by the following (see The above results were confirmed by the computer program IQ 100 [21], see Table 5.   The UGLI imperfection amplitude for the uniform member with a uniform distribution of axial force had a geometrical interpretation ( Figure 14).

Amplitudes of the Equivalent Geometrical Imperfections according to Eurocodes [7] and [18]
The values of the initial local bow imperfection / in the draft FprEN 1993-1-1:2021- 11-26 [18] were changed compared to the current Eurocode EN 1993-1-1 [7]. According to 7.3.3.1 in [18], the equivalent bow imperfection, , of the members for flexural buckling may be determined as follows: where the imperfection factor α is in Table 6 and is the relative bow imperfection according to Table 7.   Figures 19 and 20 to the values from Numerical Example 1 appears in Table 8 for buckling about axis − and in Table 9 for buckling about axis − .  It is evident in the UGLI imperfection method that safety factor cannot be removed from , , and value , cannot be used in this method. See comparisons in Tables 8 and 9.  From the comparisons in the above diagrams in Figures 19-22 for the values of bow imperfections used in the second-order theory, the results read as so: The values for the plastic cross-section verification were higher than those for the elastic ones, , > , ; The values for the buckling about the axis − were higher than those for the buckling about the axis − , , > , ;  If , was used in the second-order theory, identical results were obtained to the EM method in the ultimate state when = , . The UGLI imperfection method is based on this fact. It has long since been known that the current Eurocode [7] prescribes in 5.3.2b) much higher values than those accepted in the EM method. See, for example, the parametrical study in [22]. This could be the reason why in draft [18], the new values for almost all the combinations were lowered. The dramatic drop in the values was seen for the imperfection factor of buckling curve .
, according to draft Eurocode [18], also depends on the yield strength . Generally, the new way to calculate according to [18] gives lower values compared to those given in the current Eurocode [7]. This is valid for: all the , , values ( Figure 19); the , , values with the imperfection factors , ; and , , , , , with the imperfection factor . It is also valid for ≤ 235 MPa, except the cases for , , with the imperfection factors and .
The lower values in draft [18] are still higher than the , values used in the UGLI imperfection method. Therefore, , must not be lowered by removing from Equation (19) to obtain (8).

Conclusions
The second-order theory was used to analyze the flexural buckling of an individual member simply supported on both member ends, with a uniform double symmetric crosssection under uniform axial compressed force in an elastic state. It is well known that the equivalent member (EM) method is actually the modified second-order theory with a "hidden" amplitude , (8) of the equivalent geometrical initial imperfection, which has the shape of the elastic critical buckling mode (9).
This paper aims to show that under condition  (19) is used in the calculation; (b) the same utilization factor = 1,0 can also be obtained by the UGLI imperfection method if the design value of the imperfection amplitude , (19) is used in the calculation. It is very important to state that safety factor must not be removed from , , which was mistakenly performed in [18], where , (19) was replaced with , (8). In [7,8,17], the correct value , (19) was used in the UGLI imperfection method, as it was prescribed by its authors Chladný and Baláž in their publications [10][11][12][13][14][15][16]19,20]. Using , (8) in the UGLI imperfection method, which was incorrectly recommended in [18], leads to member resistances (and also to utility factors ) on the unsafe side, especially for very slender columns; e.g., for a relative slenderness λ = 2.0, the difference between utility factors = , and , is 41% on the unsafe side (see Figure 4). The above facts are illustrated in Numerical Example 1.
Chapter 3 informs about: (a) unknown details from historical developments of the UGLI imperfection method invented by Chladný [10,11]; (b) the best version of the UGLI imperfection method by Chladný and Baláž, which is found in [17]; and (c) illustrative Numerical Example 2, in which the column fixed at one end and simply supported on the other end was investigated. The purpose was to show that amplitudes have a geometrical interpretation ( Figure 14) for the member with a uniform cross-section and uniform axial force distribution, as previously described by Baláž in [12,19]. This invention is very important because it enables the following without almost any calculation: (a) estimating the buckling length ; (b) determining the location of the amplitude , , which is always in the middle of ; and (c) calculating the maximum bending moment due to the axial compressed force acting on the member with the equivalent geometrical imperfection from the simple formula = = , . The maximum bending moment , is located at the same point , where the amplitude , is also located. All these details, together with the shapes of the equivalent geometrical initial imperfections (46) and bending moment distributions, appear in Figures 15-17 for 17 cases. In Numerical Example 2, the hand calculation results were verified by the results of the computer program IQ 100 [21]. A perfect agreement was achieved.